Incentives and Risk Taking in Hedge Funds*

Roy Kouwenberg† AEGON Asset Management NL and Erasmus University Rotterdam

William T. Ziemba‡§

Sauder School of Business, Vancouver and Swiss Banking Institute, University of Zurich

July 17, 2003

* This research was supported by Inquire Europe. † e-mail: kouwenberg@few.eur.nl ‡ Sauder School of Business, University of British Columbia, Vancouver, B.C., Canada,

V6T 1Z2, and Swiss Banking Institute, University of Zurich, Plattenstrasse 14, CH-8032, Zurich, Switzerland, e-mail: ziemba@interchange.ubc.ca

§ Corresponding author.

1

Incentives and Risk Taking in Hedge Funds

Abstract

This paper presents a theoretical study of how incentives affect hedge fund risk and returns and an

empirical study of the performance of a large group of operating hedge funds. Most hedge fund

managers receive a flat fee plus a share of the returns above a certain benchmark. We investigate

how these features of hedge fund fees affect risk taking by the fund manager in the behavioural

framework of prospect theory. The performance related component encourages funds managers to

take excessive risk. However, risk taking is greatly reduced if a substantial amount of the

manager’s own money is in the fund as well. The empirical results indicate that hedge funds with

incentive fees have higher downside risk than funds without such a compensation contract.

Average returns though, both absolute and risk-adjusted, are significantly lower in the presence of

incentive fees.

JEL Codes: G10, G29 Keywords: hedge funds, incentive fees, optimal portfolio choice

2

As incentive fee arrangements are widespread in the hedge fund industry, it is important to

understand the impact of incentive fees on the investment strategy of hedge fund managers. This

paper presents a theoretical analysis of risk and incentives in the behavioural framework of

prospect theory, followed by an empirical study of risk taking by a large number hedge funds.

Related to our work is a paper by Carpenter (2000) that analyses the effect of incentive fees on

the optimal investment strategy of a fund manager in a continuous-time framework. Carpenter

(2000) finds that a manager with an incentive fee increases the risk of the fund’s investment

strategy if the fund value is below the benchmark specified in the incentive fee contract. This

risk taking behaviour is exactly what one expects, as the fund manager tries to increase the

value of the call option on fund value. If the fund value rises above the benchmark the manager

reduces volatility, in some cases even below the optimal volatility level of a fund without

incentive fees.

We extend the model of Carpenter (2000) along two lines: we incorporate management fees

and investments of the manager in the fund. Most fund managers charge a fixed proportion of

the fund value as management fee, in order to cover expenses and to make a living. One would

expect that management fees moderate risk taking, as negative investment returns will reduce

the future stream of income from management fees. As second important aspect of the hedge

fund industry that affects risk raking, is that most fund managers invest their own money in the

fund. This practice, known as “eating your own cooking”, helps to realign the motivation of the

fund manager with the objectives of the other investors in the fund.

The fact that hedge fund managers typically risk both their career and their own money

while managing a fund is a positive sign to outside investors. The personal involvement of the

manager, combined with a good and verifiable track record, could explain why outside investors

are willing to invest their money in hedge funds, even though investors typically receive very

limited information about hedge fund investment strategies and also possibly face poor liquidity

3

due to lock-up periods. We expect that the hedge fund manager’s own stake in the fund is an

essential factor influencing the relationship between incentives and risk taking.

We analyse the effect of incentive fees on risk taking in a continuous-time framework,

taking management fees and the manager’s own stake in the fund into account. We do not use a

standard normative utility function like HARA for the preferences of the fund manager, but the

behavioural setting of prospect theory instead. Prospect theory is a framework for decision-

making under uncertainty developed by Kahneman and Teversky (1979), based on actual

human behaviour observed in experiments. A number of recent papers argue that prospect

theory is not only relevant for small scale laboratory experiments, but can also help to explain

financial market puzzles, see Benartzi and Thaler (1995), Barberis, Huang and Santos (2001)

and Gomes (2003). Siegmann and Lucas (2002) argue that loss aversion, an important aspect of

prospect theory, can explain the non-normal return distributions of hedge funds. It is therefore

worthwhile to know how hedge fund managers driven by these preferences will react to

incentive fees.

We also derive an expression for the value of the manager’s incentive fee, as in Goetzmann,

Ingersoll, Ross (2003), and show that it can be worth more than 15% of the fund value. We

take into account the fund manager’s optimal investment strategy under prospect theory in

deriving the value of the fee. One of our findings is that loss averse hedge fund managers

increase risk taking in response to the incentive fees, regardless of whether the fund value is

above or below the benchmark. If a substantial amount of the manager’s own money in the fund

(30% or more), risk taking due to incentive fees is reduced considerably. Finally, the value of

the incentive fee option increases enormously as a result of the manager’s optimal investment

strategy, e.g. from 0.8% to 17% of initial fund value.

In the second part of the paper we complement the theoretical analysis with an empirical

study of the risk taking behaviour of a large number of hedge funds in the Zurich Hedge Fund

4

Universe (formerly known as the MAR database). We estimate cross-sectional regressions of

various risk measures on the level of the hedge fund’s incentive fee, controlling for other factors

such as investment style, management fee, size and age. Our analysis is not limited to traditional

risk measures such as volatility, but also includes downside risk, maximum drawdown,

skewness and kurtosis to capture the non-normal shape of hedge fund return distributions. The

empirical relationship between risk taking and incentives in hedge funds has been studied before

by Ackermann, McEnally and Ravenscraft (1999), Brown, Goetzmann and Park (2001),

Agarwal, Daniel and Naik (2002), but all of these papers only consider volatility as a risk

measure, while hedge fund returns are known to be non-normal (Fung and Hsieh 1997, 2001).

The paper is organized as follows. Section I develops a continuous-time framework for

deriving the fund manager’s optimal investment strategy in the framework of prospect theory,

taking into account management fees and the manager’s own investment in the fund.

Subsequently, Section I derives the value of the manager’s incentive fee contract in this setting.

Section II investigates the empirical relationship between risk taking and incentives using a

large database of hedge funds, and Section III concludes.

I. Theoretical Analysis of Incentives and Risk Taking

A. Model Setup

We consider the optimal portfolio problem of a hedge fund manager with initial wealth W(0).

The initial size of the hedge fund is Y(0). The manager owns a fraction � of the fund, with 0 � � �

1, while outside investors own the remaining portion of the assets (1-�). The management fee

equals a proportion � � 0 of fund value (1-�) Y(T) at the end of the evaluation period T (i.e. at the

beginning of the next evaluation period). The incentive fee is a percentage � � 0 of the fund’s

performance in excess of the benchmark B(T) at the end of the evaluation period T:

(1-�) � max{ Y(T) - B(T), 0 }. Similar to Carpenter (2000) we assume that the fund manager does

5

not hedge his exposure to the fund’s value with his private portfolio, i.e. his wealth outside of the

fund. For ease of exposition we set the rate of return on the private portfolio equal to the riskless

rate R(0).1 The portfolio manager’s wealth at the end of the period T is

(1) W(T) = �Y(T) + �(1 - �)Y(T) + �(1 - �)max{ Y(T) - B(T), 0 } + (1 + R(0))(W(0) - �Y(0)) .

The hedge fund manager evaluates wealth at the end of the period T with the value function of

prospect theory (Kahneman and Tverksy 1979)

(2) V(W(T)) = –A ( �(T) – W(T) 1)γ , if W(T) � �(T)

( W(T) – �(T) 2)γ , if W(T) > �(T) .

The fund manager has a threshold �(T) > 0 for separating gains and losses. The parameters 0 < �1

� 1 and 0 < �2 � 1 determine the curvature of the value function over losses and gains

respectively. The parameter A > 0 is the level of loss aversion of the hedge fund manager. In

prospect theory it is assumed that losses loom larger than gains, i.e. A > 1: the pain of a loss

exceeds the positive feeling associated with an equivalent gain.

Risky assets with prices Sk(0) for k = 1, …, K and a riskless asset with price S0(0) are

available as potential investments for the hedge fund manager. The risky asset prices follow Ito

processes with drift rate �k(t) and volatility �k(t), where t is between 0 and T, while the riskless

asset has a drift rate of r(t) and volatility of zero

(3) dS0(t) = r(t)S0(t)dt

(4) dSk(t) = �k(t)Sk(t)dt + �k(t)Sk(t)dB(t), k = 1, …, K

6

where the interest rate r(t), the vector of drift rates �(t) and the volatility matrix �(t) are adapted

processes (possibly path-dependent).

The fund manager selects a dynamic investment strategy, determined by the weights wk(t) of

risky assets k = 1, … , K in the fund, and the weight of the riskless asset w0(t), at any time t in the

continuous interval between 0 and T. For any self-financing vector of portfolio weights w(t) at

time 0 � t � T, the fund value Y(t) then follows the stochastic process (using vector notation)

(5) dY(t) = r(t)Y(t)dt + (�(t) – �r(t))'w(t)Y(t)dt + �(t)'w(t)Y(t)dB(t)

where w0(t) = 1 - �kwk(t) has been substituted and � denotes a (K×1) vector of ones.

The hedge fund manager maximizes the expectation of the value function at the end of the

evaluation period T, by choosing an optimal investment strategy for the fund using

(6) maxw(t) E[V(W(T))]

s.t. W(T) = �Y(T) + �(1- �)Y(T) + �(1- �)max{ Y(T) - B(T), 0 } + Z(T)

dY(t) = r(t)Y(t)dt + (�(t) – �r(t))'w(t)Y(t)dt + �(t)'w(t)Y(t)dB(t)

Y(t) � 0, for all 0 � t � T

where Z(T) = (1 + R(0))(W(0) - �Y(0)) denotes the fund manager’s wealth outside of the fund at T.

B. The Effect of Incentive Fees on Implicit Loss Aversion

We can analyse the effect of incentive fees on risk taking by examining the value function

V(W(T)) of the fund manager at the end of the evaluation period. Before we proceed with the

analysis we first specify the fund manager personal threshold’s �(T), separating gains from losses

7

in the value function. The hedge fund manager will only earn incentives fees if the fund value

Y(T) exceeds the benchmark value B(T) at the end of the evaluation period. Therefore, it seems

plausible that the fund value Y(T) = B(T) is the main point of focus for the manager, separating

failure from success. Just achieving the benchmark B(T) would leave the manager with the

following amount of personal wealth at the end of the year, W(T) = �B(T) + �(1-�)B(T) + Z(T).

We assume that this amount of personal wealth is the threshold that mentally separates gains from

losses for the fund manager

(7) �(T) = �B(T) + �(1- �)B(T) + Z(T) .

Given the threshold specification in equation (7), it is easy to demonstrate that the condition

W(T) � �(T) is equivalent to Y(T) � B(T) (see Appendix A for the proof). The manager will

consider fund performance below the benchmark as a loss (failure) and performance in excess of

the benchmark as a gain (success) leading to additional income from incentive fees.

Substituting the expression for W(T) in equation (1) into the value function V(W(T)), yields

(8) V(W(T)) = –A( �(T) – (� + �(1-�)) Y(T) – Z(T) 1)γ , if W(T) � �(T)

( (� + �(1-�))Y(T) + �(1-�)(Y(T) - B(T)) + Z(T) – �(T) 2)γ , if W(T) > �(T) .

Using the fact that W(T) � �(T) is equivalent to Y(T) � B(T) and substituting equation (7) for �(T)

into (8) yields the following expression for the manager’s value function

(9) V(W(T)) = –A { (� + �(1-�))( B(T) – Y(T) ) 1}γ , if Y(T) � B(T)

{ (� + (�+�)(1-�))( Y(T) – B(T) ) 2}γ , if Y(T) > B(T) .

8

We can multiply the value function by a constant, without affecting the solution of the manager’s

optimal portfolio choice problem (6). We simplify the manager’s value function back to the

standard format, multiplying V(W(T)) by ( � + (�+�)(1-�) 2) γ−

(10) V*(Y(T)) = –Â ( B(T) – Y(T) 1)γ , if Y(T) � B(T)

( Y(T) – B(T) 2)γ , if Y(T) > B(T)

where  = A ( � + �(1-�) 1)γ ⁄ ( � + (�+�)(1-�) 2)γ is the implicit level of loss aversion relevant

for the optimal portfolio choice problem of the fund manager.

Hence, under the relatively mild assumption that the manager’s personal threshold for

separating gains and losses hinges on the hedge fund’s critical level B(T) for earning incentive

fees, the manager’s objective can be reduced back to the standard prospect theory specification in

(10) as a function of fund value Y(T), with B(T) as the threshold separating gains from losses and

 as the implicit level of loss aversion. To investigate the effect of incentive fees on risk taking,

we can now examine the expression for the implicit level of loss aversion  in (11).

Proposition 1: Given 0 � � < 1, the implicit level of loss aversion  of the hedge fund manager

strictly decreases as a function of the incentive fee �, i.e. d ⁄ d� < 0.

Proposition 1 shows that an increase in the incentive fee will reduce the implicit level of loss

aversion of the hedge fund manager’s optimal portfolio choice problem. Hence, the manager of a

hedge fund with a large incentive fee should care less about investment losses than a manager

without such a fee, if the fund manager is trying to maximize the expectation of the value

function of prospect theory. Proposition 2 considers the impact of the manager’s own stake in the

fund on the implicit level of loss aversion.

9

Proposition 2: Given 0 � � � 1 and � > 0, the condition �1 = �2 is sufficient for a strictly positive

relation between the manager’s own stake in the fund � and the implicit level of loss aversion Â,

i.e. d ⁄ d� > 0.

Given �1 = �2, Proposition 2 states that a manager with a large own stake in the fund should

optimally care more about losses than a manager without such a stake. The sufficient condition

�1 = �2 means that the value function has the same curvature over gains as over losses. Tversky

and Kahneman (1992) have estimated the parameters of the value function of prospect theory

from the observed decisions made under uncertainty by a large group of people. The estimates of

Tversky and Kahneman (1992) are A = 2.25 for the average level of loss aversion and �1 = �2 =

0.88 for the curvature of the value function. As Tversky and Kahneman (1992) did not find a

significant difference between �1 and �2, the condition �1 = �2 of Proposition 2 seems plausible.

Given these estimated preference parameters, Figure 1 displays the implicit level of loss

aversion  as a function of the incentive fee for three different levels of the manager’s stake in the

fund (� = 5%, � = 20% and � = 50%). Figure 1 demonstrates that the manager’s implicit level of

loss aversion is equal to 2.25 without incentive fees (� = 0). As the incentive fee increases, the

implicit level of loss aversion of the fund manager starts to decrease, indicating that the manager

should optimally care less about losses and more about gains due to the convex compensation

structure. The negative impact of incentive fees on implicit loss aversion is mitigated to some

extent if the manager owns a substantial part the fund.

C. The Optimal Investment Strategy with Incentive Fees

We will now derive the optimal dynamic investment strategy of the fund manager. In the

previous section we reduced the value function of the fund manager back to standard format

10

V*(Y(T)), as a function of terminal fund value Y(T). Hence, the optimal portfolio choice problem

(6) is equivalent to

(11) maxw(t) E[V*(Y(T))]

s.t. dY(t) = r(t)Y(t)dt + (�(t) – �r(t))'w(t)Y(t)dt + �(t)'w(t)Y(t)dB(t)

Y(t) � 0, for all 0 � t � T .

To facilitate the solution of the optimal portfolio choice problem we assume that markets are

dynamically complete. Market completeness implies the existence of a unique state price density

(t), also known as pricing kernel, defined as

(12) d(t) = - r(t)(t)dt - κ(t)(t)dB(t), (0) = 1

where (t) = �(t)-1(t)(�(t) – �r(t)) denotes the market price of risk.

Under the assumption of complete markets, Berkelaar, Kouwenberg and Post (2003) solve the

optimal portfolio choice problem of a loss averse investor in (6) with the martingale

methodology, following Basak and Shapiro (2001). The solution is derived in two steps. First, the

optimal fund value Y*(T) is derived as a function of the pricing kernel (T) at the planning horizon

(see Proposition 3). Second, the optimal dynamic investment strategy that replicates these fund

values is derived under the assumption that the risky asset prices follow Geometric Brownian

motions and the riskless rate is constant (see Proposition 4). See Berkelaar, Kouwenberg and Post

(2003) for more details and proofs.

Proposition 3: Under the assumption of a complete market, the optimal fund value Y*(T) for the

manager at the evaluation period T, as a function of the pricing kernel, is

11

(13) Y*(T) = B(T) +

)12/(1

2

)(−

���

����

�γ

γξ Ty

, if (T) < *

0 , if (T) � *

where * solves f() = 0 with

(14) f(x) = B(T) + )21/(1

2

)21/(2

2

2 )(11 γ

γγ

γγ

γ −−

���

����

�−yx

– B(T)yx + ÂB(T)�2

and y � 0 satisfies E[ (T)Y(T) ] = (0)Y(0).

Proposition 4: Under the assumption of a complete market, Geometric Brownian motions for the

risky asset prices and a constant interest rate r(t)=r0, the optimal dynamic investment strategy

w*(t) of the fund manager as a function of the pricing kernel (t) and fund value Y(t) is

(15) w*(t)= ��

�

�

��

�

�

��

�

�

��

�

�

−+

−���

����

�+

−′ Γ

−−−−

2

*2

*2)(

)1/(1

2*

1)(1

1)((

)(

)(()()(

))(()()(

)( 2

γξ

κξφ

ξγ

κξφκσ

γdN

tT

de

tytT

detBtY

ttTr

(16) Y(t) = ))(()(

))(()( *2

)(

)21/(1

2*1

)( ξξγξ

γ

dNety

dNetB ttTr Γ−

−−���

����

�+

(17) �(t) = )(1

21

)(211 2

2

2

22

2

2 tTtTr −���

����

� −+−�

�

���

� +− κ

γγκ

γγ

(18) tT

tTrtxxd

−−−+

=κ

κξ ))(5.0())(/log()(

2

1 and 2

12 1)()(

γκ

−−

+=tT

xdxd .

where N(·) is the standard normal cumulative distribution and φ (·) is the density function.

To analyze the effect of incentive fees on the investment strategy of the fund manager, we can

use the fact that the implicit level of loss aversion  of the fund manager decreases as a function

of the incentive fee level (see Proposition 1). The next proposition shows how a decrease of Â

12

affects the optimal fund values Y*(T) at the evaluation date T.

Proposition 5: Given 0 � � < 1, an increase of the hedge fund manager’s incentive fee � will lead

to a decrease in the breakpoint * of the optimal fund value function Y*(T) and a decrease of the

Lagrange multiplier y.

Proposition 5 shows that an increase of the incentive fee makes the manager seek more payoffs in

good states of the world with low pricing kernel (due to the decrease of y) and less in bad states

(due to the decrease of *).

The effect of an increase of the incentive fee on the optimal investment strategy is illustrated

in Figure 2. For ease of exposition, we assume that there is only one risky asset, representing

equity, with a Sharpe ratio of = 0.10 and a volatility of � = 20%, and a riskless asset with r0 =

4%. The evaluation period is one year (T = 1) and the fund manager has the standard preference

parameters for the value function (A = 2.25, �1 = �2 = 0.88). The initial fund value is Y(0) = 1, the

threshold for the incentive fee is B(T) = 1, the management fee is � = 1% and the manager’s own

stake in the fund is � = 20%. Given these parameters, Figure 2 shows the optimal weight of risky

assets in the fund w*(t), as a function of fund value Y(t) at time t = 0.5. Each line in Figure 2

represents a different level of incentive fee �, ranging from 0% to 30%.

Figure 2 shows that the fund manager takes more risk in response to an increasing incentive

fee. The increase in risk is more pronounced when fund value drops below the benchmark B(T).

Due to the structure of the value function of prospect theory, a fund manager without an incentive

fee will increase risk at low fund values as well; incentive fees amplify this behaviour. Figure 3

shows the effect on the optimal investment strategy of changing the manager’s own stake in the

fund �, given an incentive fee of � = 20%. Figure 3 demonstrates that an increase of the

manager’s share in the fund can completely change risk taking. With a stake of 10% or less, the

13

manager behaves extremely risk seeking as a result of the incentive fee. However, with a stake of

30% or more, the investment strategy is very similar to the base case of 100% ownership (without

an incentive fee). Finally, Figure 4 shows the manager’s initial weight of risky assets w(0), as a

function of the incentive fee �. The different lines in Figure 4 represent different levels of the

manager’s own stake in the fund (�). Again higher incentive fees lead to increased risk taking; the

increase in risk taking is more drastic when the managers own stake in the fund is low (� 30%).

D. The Value of the Manager’s Incentive Fee Option

A typical hedge fund charges a fixed fee of 1% to 2% and an incentive fee of 20%. For hedge

fund investors it is worthwhile to know what the value of these fee arrangements is. We can use

the framework developed to determine the option value of hedge fund fees. In a complete market,

any European option with a set of payoffs X(T) at time T can be priced as follows with the pricing

kernel (T)

(19) X(0) = (0)-1 E[ (T)X(T) ]

where X(0) is the initial value of the contingent claim. The pay off of the incentive fee at time T

under the manager’s optimal strategy is X(T) = � max{ Y*(T) – B(T), 0 }. Hence, we can find the

value of the incentive fee at time 0 by calculating the expectation in (19).

Proposition 6: Under the assumption of a complete market, Geometric Brownian motions for the

risky asset prices and a constant interest rate r(t)=r0, the initial value of the incentive fee X(0),

given the fund manager’s optimal investment strategy, is

(20) X(0) = � ))(()0(

*2

)0(

)21/(1

2 ξξγ Γ

γ

dNey

−

���

����

�

14

(21) �(0) = TTr 2

2

2

22

2

2 121

211 κ

γγκ

γγ

���

����

� −+��

���

� +−

(22) T

Trxxd

κκξ )5.0())0(/ln(

)(2

1−+= and

212 1

)()(γ

κ−

+=T

xdxd .

where N(·) is the standard normal cumulative distribution.

Figure 5 plots the value of a 20% incentive fee as a function of the manager’s stake in the

fund, using the same set of parameters as in Figure 2 ( = 0.10, � = 20%, r0 = 4%, T = 1, Y(0) = 1,

B(T) = 1, � = 1% and A = 2.25, �1 = �2 = 0.88). Figure 5 shows that the value of the 20% incentive

fee ranges from 0.8% to 17% of the initial fund value, depending on the manager’s own stake in

the fund. If the manager’s stake in the fund is 100%, the manager does not care about the

incentive fee and manages the fund conservatively since it is a personal account. However, as the

manager’s stake in the fund goes to zero, the manager starts to increase the volatility of the

investment strategy in order to reap more profits from the incentive fee contract. Figure 6 shows

the optimal volatility of the fund returns Y(T)/Y(0) as a function of the manager’s stake in the

fund, given the incentive fee of 20%. Figure 6 shows that the fund manager greatly increases the

fund’s return volatility as the manager’s own stake in the fund decreases, in order to maximize the

expected payoff of the incentive fee. The increase of the value of the incentive fee due to this

change in investment behaviour is as much 2125% in this example (namely, from 0.8% to 17% of

initial fund value).

E. Other Factors Influencing Incentives and Risk Taking in the Hedge Fund Industry

We would like to point out a number of other aspects of the hedge fund industry that have not

been embedded in our theoretical model, but that also affect the relationship between incentives

and risk taking. Many incentive fee arrangements in the hedge fund industry include a high-water

mark provision, stating that losses from previous periods should be made up before any incentive

15

fees will be paid. The effect of high-water marks on risk taking is two-fold. First, if the fund value

is below the benchmark of the incentive fee arrangement at the end of the current evaluation

period, then the incentive fee option of a fund manager with a high-water mark will be ‘out-of-

the-money’ at the start of the next evaluation period. The optimal response of the fund manager to

this situation is to start the new evaluation period with a relatively risky investment strategy (see

Figure 2).

The second effect of a high-water mark provision is that it leads to a concave relationship

between fund value and the value of future incentive fee payments, which reduces risk taking. To

explain this effect, consider the strike price of the new incentive fee option issued at the

beginning of the next evaluation period (at time T). In case of losses (Y(T) < B(T)), the new option

will get strike price B(T) due to the high-water mark provision, and it will lose value rapidly as

Y(T) drops further below B(T). With gains (Y(T) > B(T)), the strike price of the new option is

adjusted upward, which puts a drag on the increase of the option value as a function of Y(T). The

result is a concave relation between fund value and the value of the new incentive fee option,

reducing the manager’s implicit level of loss aversion.

Whether high-water marks will eventually increase or decrease risk taking depends on many

parameters, such as the manager’s trade-off between current wealth and future wealth, and the

hurdle rate used for setting the benchmark B(T). An analytical study of the overall effect of high-

water marks on risk taking requires a framework with at least two performance evaluation

periods, in order to model the relationship between the manager’s current choices and the future

payoff of the incentive fee contract. Adding a second evaluation period to our continuous-time

framework will render the model analytically intractable, due to the complexity of the optimal

fund value process (16). Goetzmann, Ingersoll and Ross (2003) analyze the effect of high-water

marks analytically, however in their framework the investment strategy of the fund manager is

fixed (not optimized) and fees are earned continuously instead of periodically.

16

Fund flows are another important factor affecting risk taking in a multi-period setting.

Agarwal, Daniel and Naik (2002) find that the relationship between hedge fund performance and

subsequent fund flows is convex, just like in the mutual fund industry (Chevalier and Ellison

1997, Sirri and Tufano 1998). Outperforming hedge funds attract significant amounts of new

money, while withdrawals after poor performance are relatively small. The convex flow-

performance relation creates an incentive for fund managers to increase risk taking, especially

after poor performance. Other aspects that might affect risk taking are peer group pressure and the

expectations of investors about the appropriate risk level of hedge funds. For example, hedge

funds within the market-neutral style group have on average displayed relatively low return

volatility in the past. Therefore, a market-neutral hedge fund with highly volatile returns will

probably be looked upon rather suspiciously by investors, as the fund does not conform to the

characteristic of its peer group. Increased risk taking could therefore lead to fund outflows

(Agarwal, Daniel and Naik 2002 report evidence for this relationship).

All in all, the number of factors that can influence the relationship between risk taking and

incentives in the hedge fund industry is rather overwhelming. It is not the aim of this paper to

include every aspect of the hedge fund industry in a theoretical analysis of risk taking and

incentives, as the model is intended to be a simplified image of reality, and not reality itself.

Instead, in the next section we proceed with an empirical investigation, in order to see whether

hedge funds with incentive fees indeed take on more risk than funds without such an

arrangement.

II. Empirical Analysis of Incentives and Risk Taking in Hedge Funds

A. Hedge Fund Data

For our empirical investigation we use the Zurich Hedge Fund Universe, formerly known

as the MAR hedge fund database, provided by Zurich Capital Markets. The database includes

17

a large number of funds that have disappeared over the years, which reduces the impact of

survivorship bias. The data starts in January 1977 and ends in November 2000. Overall there

are 2078 hedge funds in the database and 536 fund of funds. We will analyse the hedge fund

data from January 1995 to November 2000 since the database keeps track of funds that

disappear starting January 1995. The return data is net of management fees and net of

incentive fees.

The hedge funds in the database are classified into eight different investment styles by the

provider: Event-Driven, Market Neutral, Global Macro, Global International, Global Emerging,

Global Established, Sector and Short-Sellers. We will merge the styles Global International,

Global Established and Global Macro into one group, denoted Global Funds, as these three styles

have similar investment style descriptions. We treat the Global Emerging funds as a separate

category, denoted Emerging Markets, as the funds within this style are often unable to short

securities and emerging market funds have quite different return characteristics compared to the

other global funds.

Table 1 provides some descriptive statistics of the funds in the database. The table

distinguishes between funds that were still in the database in November 2000 (alive) and funds

that dropped out (dead) and between individual hedge funds and fund of funds. The median

incentive fee for hedge funds is 20%. An incentive fee of 20% seems to be the industry standard,

as 71.4% of the funds use it. Only 8.5% of all hedge funds do not charge an incentive fee. The

median management fee is 1%. The majority of funds (71.5%) charge a fee between 0.5% and

1.5%, while only 4.2% of the funds do not charge a management fee. On top of this, an investor

in fund of funds has to pay fees to the fund of fund manager. On average fund of funds charge

slightly lower fees than individual hedge funds, although the median incentive fee is still 20%

(dead and alive funds combined). Only 6.2% of fund of funds do not charge an incentive fee. The

median management fee of fund of funds is 1%. The funds in the database are relatively young,

18

with an average age of 4 years for living funds and 2.6 years for dead funds (same for hedge funds

and fund of funds). The relatively young age of the funds has to do with the rapid growth of the

hedge fund industry over the period 1995-2000. For a study of the performance of the funds in the

database over this period see Kouwenberg (2003).

B. Incentives and Risk Taking in Hedge Funds: Empirical Results

Empirical studies of incentives and risk taking in the literature typically test whether funds

with poor performance in the first half of the year increase risk in the second half of the year, (see

e.g.. Brown, Harlow and Starks 1996, Chevalier and Ellison 1997 and Brown, Goetzmann and

Park 2001). The idea behind this approach is that funds with an incentive fee, or facing a convex

performance-flow relationship, will increase risk after bad performance in the first half of the year

in order to increase the value of their out-of-the-money call option on fund value. Considered

within the context of the prospect theory framework applied in this paper, such a test is less

meaningful. Loss averse fund managers will always increase risk as their wealth drops below the

threshold, regardless of incentive fees (see Figure 2). A more distinguishing effect of incentive

fees within the prospect theory framework is that incentives reduce implicit loss aversion and lead

to increased risk taking across the board, even at the start of the evaluation period (see Figure 4).

We will therefore test if the risk of hedge funds returns increases as a function of the fund’s

incentive fee.

Hedge fund returns are well known to be non-normal due to the dynamic investment

strategies of the funds (see Fung and Hsieh 1997, 2001 and Mitchell and Pulvino 2001). Still,

empirical studies of the relationship between risk taking and incentives in hedge funds only

consider volatility as a risk measure (Ackermann, McEnally and Ravenscraft 1999, Brown,

Goetzmann and Park 2001 and Agarwal, Daniel and Naik 2002), even though volatility can not

fully capture the non-normal shape of hedge fund return distributions. To address this problem,

19

we will focus on non-symmetrical risk measures, namely the 1st downside moment and

maximum drawdown, as well as the skewness and kurtosis of hedge fund returns. The 1st

downside (upside) moment is defined as the conditional expectation of the fund returns below

(above) the risk free rate. Maximum drawdown is defined as the worst performance among all

runs of consecutive negative returns.

Table 2 shows the cross-sectional average of ten different risk and return measures of the

hedge funds in the database, conditional on the level of the incentive fee. The risk measures are

volatility, 1st downside moment (relative to the risk free rate), maximum drawdown, skewness

and kurtosis. The return measures are the fund’s mean return and 1st upside moment. Finally,

Table 2 lists three risk-adjusted performance measures, namely the Sharpe ratio, Jensen’s alpha

and the gain-loss ratio. The gain-loss ratio is defined as the ratio of the 1st upside moment to the

1st downside moment. Berkelaar, Kouwenberg and Post (2003) demonstrate that the gain-loss

ratio can be interpreted as a measure of the investor’s implicit level of loss aversion. The last

column of Table 2 displays the p-value of an ANOVA-test for differences in means between the

incentive fee groups.

The first row of Table 2 shows that hedge funds without incentive fee, on average, have

considerably higher mean returns than funds that do charge an incentive fee (means are

significantly different between groups). Apparently funds with an incentive fee cannot make up

for the costs of the fee. We do not find statistically significant evidence that incentive fees lead to

drastic changes in average volatility, 1st downside moment and maximum drawdown of hedge

funds. We do find significant differences in average skewness and kurtosis between incentive fee

groups. However, the latter finding seems to be caused mainly by the relatively small group of

funds with an incentive fee in excess of 20%.

When we examine the results for the three risk-adjusted performance measures, Sharpe ratio,

alpha and gain-loss ratio, we find significant differences between incentive fee groups. Funds

20

without an incentive fee achieve the best risk-adjusted performance on average, while funds

charging a below average incentive fee have relatively poor performance. Overall, we conclude

from Table 2 that incentive fees reduce the mean return and risk-adjusted performance of funds,

while the effects on risk are not very clear-cut. We have also analysed the data after correcting for

differences in investment styles by measuring deviations from the average in each style group, but

the conclusions do not change materially (results available on request from the authors).

To control for other hedge fund characteristics such as fund size, age, management fee and

investment style group, we also estimate the following cross-sectional regression model for the

hedge fund risk and return measures

(23) ai = � =Hh ihd1 + ifi + mfi + navi + agei + �i ,

with i = 1, …, I and �i ~ N(0, ) independently normally distributed,

where ai denotes the cross-sectional hedge fund statistic under consideration of fund i = 1, …, I,

dih is a dummy which equals one if fund i belongs to hedge fund style h = 1, ..., H and zero

otherwise, ifi is the incentive fee, mfi the management fee, navi is the mean net asset value of the

fund and agei is the number of years that the fund is in the database.

Table 3 reports the cross-sectional regression results. Columns 2 to 6 in Table 3, denoted by

Regression A, refer to regression model (23) above; columns 7 to 11, denoted by Regression B,

refer to a slightly modified version of the model, which uses a dummy variable for the incentive

fee and a dummy for the management fee; the dummy variables are one if a fee is charged and

zero otherwise. We do not report the estimated hedge fund style dummies dih in Table 3 to save

space. If we concentrate on the results for the incentive fee variable in Table 3, we find that funds

with higher fees earn significantly lower mean returns. The only other significant effect of

incentive fees is a reduction of Sharpe ratios and alphas (only in Regression B, with incentive fee

21

dummies). There is no significant effect of incentive fees on any of the five risk measures at the

5% confidence level. However, we do see an economically relevant increase of the 1st downside

moment and the maximum drawdown due to incentive fees, as the estimated coefficients are quite

substantial. Moreover, the increase in the 1st downside moment is significant at the 10% level in

both regressions. We conclude that there is some evidence for increased risk taking due to

incentive fees if we focus on downside risk instead of volatility as a risk measure.

C. Incentives and Risk Taking in Fund of Funds: Empirical Results

We now repeat the empirical analysis for the fund of funds in the database. Table 4 displays

the cross-sectional average of the ten risk and return measures, conditional on the level of the

incentive fee. We use three incentive fee groups instead of four, due to the relatively small

number of fund of funds (403 in total). Again we find significant differences between the average

mean returns of the incentive fee groups. Fund of funds with high fees earn higher returns on

average. The 1st upside moment is also significantly different across groups and larger for fund of

funds with fatter fees. We do not find significant differences in the five risk measures between

groups. The three risk-adjusted performance measures, Sharpe ratio, alpha and gain-loss ratio, are

significantly different across groups and relatively large for fund of funds with high fees (� 20%).

Table 5 contains the estimation results of the cross-sectional regression model (23) for fund of

funds.2 The coefficient of the incentive fee variable is significantly positive in the cross-sectional

regression on the 1st upside moment, volatility, maximum drawdown and gain-loss ratio (at the

5% level). There is an economically relevant positive impact on the mean return, 1st downside

moment, skewness and Sharpe ratio as well, based on the magnitude of the estimated coefficients.

Hence, for the fund of funds in the database we find that higher incentive fees are linked to

increased upside potential and increased risk taking. Risk-adjusted returns increase as well, so

investors seem to be better of with fund of funds that charge higher incentive fees. In the case of

22

management fees, Table 5 shows that they are a drag on performance: higher fees significantly

reduce average returns, Sharpe ratios and alphas.

A potential explanation for the positive relationship between incentive fees and (risk-adjusted)

returns in Table 4 and 5 might be that fund of fund managers with incentive fees opt for a more

risky basket of hedge funds to increase the value of their call option on fund value, leading to

more upside return potential and more risk as well. The fund of fund managers themselves might

argue that funds with better manager selection skills generate higher returns and are therefore able

to charge higher incentive fees. A weak point of the latter story is that it does not explain why

fund of fund managers with better skills have more risky returns on average as well; the skill

advantage should allow good managers to achieve better returns, while taking less risk.

23

III. Conclusions

In this paper we analyse the relationship between incentives and risk taking in the hedge

fund industry. We use prospect theory to model the hedge fund manager’s behaviour and derive

the optimal investment strategy for a manager in charge of a fund with an incentive fee

arrangement. We find that incentive fees reduce the manager’s implicit level of loss aversion,

leading to increased risk taking. However, if the manager’s own stake in the fund is substantial

(e.g. > 30%), risk taking will be reduced considerably. We also derive an expression for the

option value of the incentive fee arrangement, taking into account the manager’s optimal

investment strategy. We show that the fund manager increases the value of the incentive option

by increasing the volatility of fund returns.

In the second part of the paper we examine empirically whether hedge fund managers with

incentive fees indeed take more risk in practice, using the Zurich Hedge Fund Universe

(formerly known as the MAR database) in the period January 1995 to November 2000. The

cross-sectional analysis mainly shows that hedge funds with incentive fees have significantly

lower mean returns (net of fees) and worse risk-adjusted performance. There is no significant

effect on volatility, but the 1st downside moment of returns increases substantially in the

presence of incentive fees (significant at the 10% level). Our results illustrate the importance of

using downside risk measures, given the non-normality of hedge funds returns.

Funds of funds charging higher incentive fees have more risky and higher returns on

average. Hence, it seems that fund of funds take more risk in response to incentive fees. It seems

unlikely that fund of fund managers with higher incentive fees are more skilful, as that story

does not explain why risk taking increases as well as a function of incentive fees.

24

REFERENCES

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Risk, Return and Incentives, Journal of Finance, vol. 54, 833-874.

Agarwal, V., Daniel, N.D. and N.Y. Naik, 2002, Determinants of Money-Flow and Risk-taking

Behavior in the Hegde Fund Industry, Working paper, London Business School.

Barberis, N., Huang, M. and Santos, T, 2001, Prospect Theory and Asset Prices, Quarterly

Journal of Economics, vol. 116, 1-53.

Basak, S., Pavlova, A. and A. Shapiro, 2003, Offsetting the Incentives: Risk Shifting, and

Benefits of Benchmarking in Money Management, Working paper, MIT.

Basak, S. and A Shapiro, 2001, Value-at-Risk Based Risk Management: Optimal Policies and

Asset Prices, Review of Financial Studies, vol. 14, 371-405.

Benartzi, S. and Thaler, R., 1995, Myopic Loss Aversion and the Equity Premium Puzzle,

Quarterly Journal of Economics, vol. 110, 73-92.

Berkelaar, A.B., Kouwenberg, R. and G.T. Post, 2003, Optimal Portfolio Choice under Loss

Aversion, Working paper, Erasmus University Rotterdam.

Brown, K.C., Harlow, W.V. and L.T. Starks, 1996, Of Tournaments and Temptations: An

Analysis of Managerial Incentives in the Mutual Fund Industry, Journal of Finance,

vol. 51, 85-109.

Brown, S.J., Goetzman, W.N. and J. Park, 2001, Careers and Survival: Competition and Risk in

the Hedge Fund and CTA Industry, Journal of Finance, vol. 56, 1869-1886.

Carpenter, J., 2000, Does Option Compensation Increase Managerial Risk Appetite?, Journal of

Finance, vol. 55, 2311-2331.

Chevalier. J. and G. Ellison, 1997, Risk Taking by Mutual Funds as a Response to Incentives,

Journal of Political Economy, vol. 105, 1167-1200.

25

Elton, J.E., Gruber, M.J., and C.R. Blake, 2003, Incentive Fees and Mutual Funds, Journal of

Finance, vol. 58, 779-804.��

Fung, W. and D.A. Hsieh, 1997, Empirical Characteristics of Dynamic Trading Strategies: The

Case of Hedge Funds, Review of Financial Studies, vol. 10, 275-302.

Fung, W. and D.A. Hsieh, 2001, The Risk in Hedge Fund Strategies: Theory and Evidence from

Trend Followers, Review of Financial Studies, vol. 14, 313-341.

Goetzmann, W.N., Ingersoll, J.E. and S.A. Ross, 2003, High-Water Marks and Hedge Fund

Management Contracts, Journal of Finance, forthcoming.

Gomes, F., 2000, Portfolio Choice and Trading Volume with Loss Averse Investors, Journal of

Business, forthcoming.

Kahneman, D.H. and A. Tversky, 1979, Prospect Theory: An Analysis of Decision Making

under Risk, Econometrica, vol. 47, 263-290.

Kouwenberg, R., 2003, Do Hedge Funds Add Value to a Passive Portfolio: Correcting for Non-

Normal Returns and Disappearing Funds, Journal of Asset Management, vol. 3, 361-382.

Mitchell, M. and T. Pulvino, 2001, Characteristics of Risk and Return in Risk Arbitrage,

Journal of Finance, vol. 56, 2135-2175.

Siegmann, A. and A. Lucas, 2002, Explaining Hedge Fund Investment Styles by Loss Aversion,

Working paper, Vrije Universiteit Amsterdam.

Sirri, E.R. and P. Tufano, 1998, Costly Search and Mutual Fund Flows, Journal of Finance,

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Representation of Uncertainty, Journal of Risk and Uncertainty, vol. 5, 297-323.

26

APPENDIX A

Proof of the equivalence of W(T) � �(T) and Y(T) � B(T)

Substituting (1) for W(T) and (7) for �(T), W(T) � �(T) is equivalent to

(A1) �Y(T) + �(1 - �)Y(T) + �(1 - �)max{ Y(T) - B(T), 0 } + Z(T) � �B(T) + �(1- �)B(T) + Z(T).

Rearranging terms to the left-hand-side, it follows that (A1) is equivalent to (A2)

(A2) �(Y(T)- B(T)) + �(1 - �) (Y(T)- B(T)) + �(1 - �)max{ Y(T) - B(T), 0 } � 0.

Finally, it is easy to see that (A2) holds if and only if Y(T) � B(T).

Proof of Proposition 1

The manager’s implicit level of loss aversion is

(A3) Â = A ( � + �(1-�) 1)γ ⁄ ( � + (�+�)(1-�) 2)γ.

Differentiating  with respect to � yields

(A4) d / d� = -�2(1-�) A ( � + �(1-�) 1)γ ⁄ ( � + (�+�)(1-�) 12) +γ .

Given 0 � � < 1, A > 0, � � 0, � � 0 and 0 < �2 � 1, the inequality d / d� < 0 holds.

27

Proof of Proposition 2

The manager’s implicit level of loss aversion is given in (A3).

Differentiating  with respect to � yields

(A5) d / d� =  ( �1(1-�)(� + �(1-�))-1 - �2(1-�-�)(� + (�+�)(1-�))-1 ) .

Note that d / d� > 0 is equivalent to

(A6) �1(1-�)(� + (�+�)(1-�)) > �2(1-�-�)(� + �(1-�)) .

Given 0 � � < 1, � � 0, � > 0, the condition �1 = �2 is sufficient for d / d� > 0.

Proof of Proposition 3

See Berkelaar, Post and Kouwenberg (2003).

Proof of Proposition 4

See Berkelaar, Post and Kouwenberg (2003).

Proof of Proposition 5

An increase of the manager’s incentive fee � leads to a reduction of the implicit level of loss

aversion  (see Proposition 1). Berkelaar, Post and Kouwenberg (2003) prove that a decrease of Â

leads to a decrease in the breakpoint * of the optimal fund value function Y*(T) and a decrease of

the Lagrange multiplier y.

28

Proof of Proposition 6

We substitute the pay off of the incentive fee contract X(T) into equation (19) to yield (A7):

(A7) X(0) = (0)-1 E[ (T) � max{ Y*(T) – B(T), 0 } ]

Equation (A7) can be simplified by substituting the manager’s optimal fund value function Y*(T)

(A8) X(0) = )()(

)(*

0

)12/(1

2

TdTy

T ξγξβξ

ξ γ

�−

���

����

� .

Under the assumption of a complete market, a constant risk free rate and Geometric Brownian

motions for the asset prices, the pricing kernel (T) at time T is log-normally distributed. Using

this fact, the expression in (20) for X(0) follows from solving the integral (A8).

29

Table 1 Characteristics of the hedge fund data Descriptive statistics of the hedge funds in the database from January 1995 to November 2000 are displayed. The cross-sectional mean, median and standard deviation of the incentive fee, the management fee, the net asset value and the age of the fund are listed respectively. Alive funds were still in the database in November 2000, while dead funds dropped out before this date. We exclude funds that do not report fees.

Hedge funds Fund of funds Alive Dead Alive Dead Incentive fee Mean

Median Std. Dev.

18.2 20.0 6.2

17.9 20.0 7.0

16.7 20.0 5.2

13.5 15.0 7.4

Management fee Mean Median Std. Dev.

1.22 1.00 0.52

1.30 1.00 0.64

1.04 1.20 0.75

1.12 1.00 0.70

Net Asset Value Mean Median Std. Dev.

2,361 1,190 9,941

2,260 1,015

11,107

1,319 1,121 2,438

1,004 158

21,254

Age (years) Mean Median Std. Dev.

3.81 3.92 1.81

2.64 2.42 1.60

4.17 4.58 1.85

2.58 2.42 1.53

Number of funds 772 470 307 144

30

Table 2 Hedge fund risk and returns conditional on incentive fee The average cross-sectional risk and returns of the hedge funds in the database from January 1995 to November 2000 are displayed, conditional on the incentive fee charged by the fund. The sample includes both funds that were still in the database in November 2000 and funds that dropped out before this date. Only funds with at least 12 monthly observations are included. The last column denotes the p-value of an ANOVA-test for equality of the means of the four incentive fee groups. †As outliers influence the mean of the gain-loss ratio heavily, we use the median instead and report the p-value of the Kruksal-Wallis test for differences in the median in the last column.

Overall Mean conditional on incentive fee mean if=0 0<if<20 if=20 if >20 p-value Average return 15.4 23.7

13.3 15.2

11.8 0.0%

1st Upside moment 30.6

37.1 31.2 30.4 23.2 0.4%

Volatility 20.0

21.4 21.0 19.8 17.9 45.6%

1st Downside moment 20.2

18.4 23.0 20.2 16.5 16.3%

Maximum drawdown 23.0

19.4 26.4 22.8 22.9 19.8%

Skewness -0.49

-0.30 -0.73 -0.42 -1.01 0.0%

Kurtosis 6.61

5.93 6.65 6.42 9.29 0.1%

Sharpe ratio 0.85

1.21 0.42 0.88 0.97 0.0%

Alpha 0.36

0.97 -0.02 0.37 0.29 0.0%

Gain-Loss Ratio† 1.73

2.23 1.40 1.75 1.58 0.0%

Number of funds 1114 93 150 789 82

31

Table 3 Cross-sectional regressions of hedge fund risk and returns The results of a cross-sectional regression are displayed of the risk and returns of the hedge funds in the database from January 1995 to November 2000 on the level of the fund’s incentive fee (if), management fee (mf), mean net asset value (nav), the age of the fund (age) and dummies for the hedge fund style groups (not reported below). The explanatory variables if, mf and nav are scaled by the median value of the series, while age is measured in years, in order to facilitate interpretation of the coefficients. In Regression B dummy variables for the incentive fee (ifd) and management fee (mfd) are used, instead of actual values as in Regression A. Only funds with at least 12 monthly observations are included. White heteroskedasticity-consistent p-values are reported in brackets below the estimates. *Denotes significance at the 5% level.

Regression A Regression B if mf nav age R2 ifd mfd nav age R2 Average return -3.57*

(0.05) -0.11 (0.90)

0.01 (0.77)

0.46 (0.24)

13% -7.03* (0.00)

1.87 (0.29)

0.01 (0.78)

0.61 (0.12)

13%

1st Upside moment -0.86 (0.72)

1.92 (0.13)

-0.04 (0.42)

-1.19* (0.01)

26% -3.62 (0.22)

-2.56 (0.44)

-0.04 (0.43)

-1.25* (0.00)

25%

Volatility 1.18 (0.42)

2.80* (0.01)

-0.03 (0.27)

-0.87* (0.00)

23% 0.05 (0.98)

-2.05 (0.45)

-0.03 (0.29)

-1.01* (0.00)

22%

1st Downside moment 2.69 (0.10)

2.01 (0.10)

-0.05 (0.13)

-1.67* (0.00)

19% 3.39 (0.07)

-4.42 (0.23)

-0.04 (0.14)

-1.88*

(0.00) 19%

Maximum drawdown 2.78 (0.13)

4.23* (0.01)

-0.04 (0.23)

1.44* (0.00)

18% 2.61 (0.21)

-4.90 (0.28)

-0.04 (0.25)

1.15* (0.01)

17%

Skewness 0.03 (0.80)

0.02 (0.88)

-0.00 (0.93)

-0.15*

(0.00) 6% -0.03

(0.85) 0.17

(0.31) -0.00 (0.92)

-0.14* (0.00)

7%

Kurtosis -0.15 (0.78)

1.35* (0.01)

0.02 (0.13)

1.02* (0.00)

12% -0.37 (0.57)

1.19 (0.07)

0.02 (0.15)

0.98*

(0.00) 11%

Sharpe ratio -0.10 (0.38)

-0.08 (0.31)

0.00 (0.08)

-0.01 (0.77)

8% -0.35* (0.01)

-0.40 (0.20)

0.00 (0.06)

-0.00 (0.88)

9%

Alpha -0.19 (0.21)

-0.00 (0.97)

0.00 (0.98)

0.01 (0.75)

9% -0.51* (0.00)

0.20 (0.22)

0.00 (0.99)

0.02 (0.49)

10%

Gain-Loss Ratio 1.98 (0.17)

0.06 (0.97)

0.02 (0.50)

-1.58* (0.03)

2% 2.67 (0.15)

-0.13 (0.97)

0.02 (0.49)

-1.62*

(0.04) 2%

32

Table 4 Fund of funds risk and returns conditional on incentive fee The average cross-sectional risk and returns of the fund of funds in the database from January 1995 to November 2000 are displayed, conditional on the incentive fee charged. Only fund of funds with at least 12 monthly observations are included. The last column denotes the p-value of an ANOVA-test for equality of the means of the four incentive fee groups. †As outliers influence the mean of the gain-loss ratio heavily, we use the median instead and report the p-value of the Kruksal-Wallis test for differences in the median in the last column.

Overall

Mean conditional on incentive fee (if)

Mean if=0 0<if<20 if�20 p-value Average return 11.8 10.5

10.0 12.9

1.8%

1st Upside moment 18.0

15.5

16.1 19.3 0.9%

Volatility 11.3

9.1 10.5 11.9 16.3%

1st Downside moment 11.3

10.0 11.2 11.5 87.8%

Maximum drawdown 14.4

9.2 14.0 15.0 37.0%

Skewness -0.48

-0.65 -0.55 -0.42 50.1%

Kurtosis 6.13

5.04

6.20 6.15 61.2%

Sharpe ratio 0.92

0.84 0.71 1.05 1.1%

Alpha 0.18

0.11 0.02 0.28 4.8%

Gain-Loss Ratio† 1.88

1.72 1.76 2.04 2.4%

Number of funds 403 16 140 247

33

Table 5 Cross-sectional regressions of fund of funds risk and returns The results of a cross-sectional regression are displayed of the risk and returns of the fund of funds in the database from January 1995 to November 2000 on the level of the fund’s incentive fee (if), management fee (mf), mean net asset value (nav) and the age of the fund (age). The explanatory variables if, mf and nav are scaled by the median value of the series, while age is measured in years, in order to facilitate interpretation of the coefficients. Only funds with at least 12 monthly observations are included. White heteroskedasticity-consistent p-values are reported below the estimates. *Denotes significance at the 5% level. if mf nav age R2 Average return 2.36

(0.16) -2.02* (0.01)

0.18 (0.18)

0.15 (0.69)

3%

1st Upside moment 5.62* (0.00)

0.29 (0.68)

-0.15 (0.41)

-0.14 (0.65)

2%

Volatility 3.66* (0.00)

1.41* (0.01)

-0.20 (0.14)

-0.10 (0.67)

3%

1st Downside moment 3.22 (0.06)

2.30* (0.00)

-0.33 (0.06)

-0.32 (0.41)

3%

Maximum drawdown 4.92* (0.03)

4.26* (0.00)

-0.32 (0.12)

1.46* (0.00)

6%

Skewness 0.29 (0.14)

-0.15 (0.14)

-0.05* (0.03)

-0.12*

(0.00) 5%

Kurtosis -0.13 (0.86)

0.45 (0.19)

0.13 (0.22)

0.74* (0.00)

9%

Sharpe ratio 0.32 (0.07)

-0.27* (0.00)

0.01 (0.39)

-0.02 (0.55)

5%

Alpha 0.15 (0.33)

-0.21* (0.01)

0.00 (0.78)

0.01 (0.87)

3%

Gain-Loss Ratio 4.10* (0.03)

0.43 (0.76)

0.10 (0.48)

-0.26 (0.63)

1%

34

Figure 1 Implicit level of loss aversion as a function of incentive feewith fixed fee of �=1%, lines for different levels of the manager's stake in the fund (�)

���

� ��

� ��

� ��

���� � �� � � ��� � �� � � ���� � � �� � � � ��� � � �� � � ���� � � �� � � � ���

Incentive fee % (�)

Impl

icit

leve

l of l

oss

aver

sion

(Â)

� � �

� � ��

� � ��

Figure 2 Optimal weight of stocks as a function of fund valuemanager's stake in the fund �=20%, lines for different levels of the incentive fee (�)

��

� ���

� ���

� ���

� ���

��� � ��� � ��� � �� � �� � � ��� � ��� � �� � � �� � � �� � � �� �

Fund value Y (t ) at time t = 0.5

Opt

imal

wei

ght o

f sto

cks � = 30%

� = 25%

� = 20%

� = 15%

� = 0%

35

Figure 3 Optimal weight of stocks as a function of fund valueincentive fee of �=20%, lines for different levels of the manager's stake in the fund (�)

��

� ���

� ���

� ���

��� � ��� � ��� � �� � �� � � ��� � ��� � �� � � �� � � �� � � �� �

Fund value Y (t ) at time t = 0.5

Opt

imal

wei

ght o

f sto

cks

� = 100%

� = 30%

� = 20%

� = 15%

� = 10%

Figure 4 Initial weight of stocks as a function of the incentive feelines for different levels of the manager's stake in the fund (�)

0%

100%

200%

300%

�� � �� � � � � �� � � �� � � �� � � � � � � �� � � �� � � �� � � � � � � �� � � ��

Incentive fee (�)

Initi

al w

eigh

t of s

tock

s

� = 100%

� = 30%

� = 20%

� = 10%

36

Figure 5 Option value of 20% incentive fee, as a function of the manager's stake in the fund (�)

0.00

0.05

0.10

0.15

0.20

�� � �� � �� � �� � �� � �� � �� � �� � �� �� � ���

Manager's stake in the fund (�)

Val

ue o

f 20

% in

cent

ive

fee

(fun

d va

lue

= 1)

Figure 6 Optimal volatility of fund returns with incentive fee of 20%, as a function of the manager stake in the fund (�)

0%

100%

200%

300%

400%

�� � � � �� � � � � �� � � � � �� � � � � �� � � � � ��

Manager's stake in the fund (�)

Vol

atili

ty o

f hed

ge fu

nd re

turn

s

37

1 All of the following results are also valid for the case where the private portfolio’s return

R(0) is stochastic.

2 In this case we do not report additional results for a regression with incentive fee dummies

and management fee dummies as there are only a few funds with zero incentive fees, leading

to a lack of statistical power.